Question of the Day - 04 August 2020

[Editor's Note: Michael "Wizard of Odds" Shackleford, author of Gambling 102, now in its second edition, graciously answered this question.] 

As a basis of comparison, the probability of a royal flush in conventional video poker ranges from about 1 in 40,000 to 45,000, assuming optimal strategy.  Here are some exact figures for some randomly selected games:

9-6 Jacks or better:  1 in 40,391

25-15-9-4-4-3 (Illinois) Deuces Wild:  1 in 43,423

9-7 Triple Double Bonus:  1 in 45,358

For royal-hungry players, the probability goes up significantly in Chase the Royal.  This is an early video poker variant where the player may exchange a deal pair of face cards for three to a royal flush on the deal.  To make it a good value to trade, the game bumps up the win on the straight and flush, if you switch.  The exact royal probability will depend on the base game and pay table.  The probability is maximized with 8-5 Triple Bonus Chase the Royal at 1 in 9,151.  This includes both Royals on the draw, which pay 800 for 1, at a frequency of 1 in 9,282 and on the deal, which pay 2000 for 1, with a frequency of 1 in 649,773.

However, it gets even better if you consider games in which the player must pay a fee to enable a feature.  In Draw Poker with Dream Card (not to be confused with Dream Card Poker), the player often gets the card of his dreams (assuming its a mathematician doing the dreaming) as the fifth card on the deal. The Dream Card probability of maximized in Jacks or Better, at 50.5%.  In 9-6 Jacks or Better, the overall royal frequency is 1 in 8,105. Keep in mind that with the fee to enable the feature, the effective win on a royal drops to 400 for 1.

As long as I mention Dream Card Poker (a different game than Draw Poker with Dream Card) the royal frequency in that game is not as high.  It seems to be highest in 11-8-6 Jacks or Better at 1 in 15,034. 

This answer does not include Movin' On Up Poker, which is an old and obscure video poker game where the player gets two or three draws, instead of one.  I don't know the royal frequency in that game, but in Triple Draw, in which the player must pay a fee equal to five times his base wager to enable the extra two draws, I roughly estimate it to be about 1 in 4,000. 

In conclusion, if you don't count games where the player must pay an extra fee to enable some kind of gimmick, my answer is Chase the royal.